The conserved quantity arising from a rotational invariance. Combine with rotational-dynamics for the classical mechanics approach and quantum-mechanics for the QM interpretation

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38 views

Why does the kinetic energy of a particle moving in circular motion increase when the turn radius decreases and no torque is acting?

Why does the kinetic energy of a particle moving in circular motion increase when the turn radius decreases and there is no torque acting? E.g. if a planet is rotating about its axis and it shrinks to ...
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1answer
109 views

Why does the magnitude of linear momentum of a particle in circular motion change with radius? [duplicate]

My problem is with linear momentum of a particle in circular motion. If we imagine a particle moving around a circle, if there are no torques acting, then we can say its angular momentum is conserved, ...
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0answers
43 views

Spin Orbital Coupling matrix in p-orbital basis

So I have the following Hamiltonian inherited from atomic Physics: $H_{SOC}=\alpha \vec{L}\cdot \vec{S}=\frac{\alpha}{2}(L^{+}\sigma^{+}+L^{-}\sigma^{-}+ L^{z}\sigma^{z})$ Where L is the angular ...
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1answer
19 views

Spheres collide with merry-go-round [closed]

Four spheres, with uniform densities $\rho_1, \rho_2, \rho_3, \rho_4$ and radii $r_1, r_2, r_3, r_4$, respectively, roll without slipping with constant velocities $v_1, v_2, v_3, v_4$ along tracks ...
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1answer
38 views

Angular momentum in rolling about the point of contact

A cylinder of mass 5 kg and radius 10 cm is moving on a horizontal surface with velocity of centre of mass 5 m/s towards right and angular speed 10 rad/s (clockwise) . Find the angular momentum of the ...
2
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1answer
57 views

Parallel axis theorem and Koenig theorem for angular momentum

Are the parallel axis theorem and the Koenig theorem for angular momentum linked with each other in rigid body dynamics? The parallel axis theorem states that $$I_{z}=I_{cm}+ma^2$$ Koenig theorem ...
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35 views

Relative angular momentum?

Let there be a point $P$. A point $C$ is located at a radius vector $r$ from $P$. $C$ is the centre of mass of a rigid body. The rigid body is rotating with an angular velocity $\omega$ about an axis $...
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1answer
27 views

Derivative of angular momentum of rigid body

I found this equation that describes the change in angular momentum $\vec{L}$ of a rigid body rotating about a fixed point $O$. $I_o$ is the moment of inertia of the body with respect to the axis of ...
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1answer
22 views

Calculation of support reaction in rigid body rotation and collisions

I can't understand the logic behind the calculation of torques exerted by supports in rigid body motion, especially rotation. The equation of angular momentum is $$\vec{\tau}=\frac{d\vec{L}}{dt}\tag{...
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1answer
48 views

Why is the center of mass frame always used in rigid body dynamics?

In most of the cases the center of mass is chosen for rigid body motion description, but this is not an obliged choice, since the motion of any point $P$ of the rigid body can be seen as the ...
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2answers
48 views

Disk let free to rotate

A rigid body moving with no constraints, in particular rotating, will rotate necessarily about a principal axis of inertia. I thought that the reason of this is that otherwise, the angular momentum $\...
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1answer
50 views

Principal axes of inertia of a compound pendulum

I am confused about principal axes of inertia. Consider the compount pendulum in the picture, made of a rectangular plate. I oscillates about a horizontal axis $\hat{a}$ passing through $A$. The ...
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34 views

Spin Orbit Coupling Hamiltonians

I am really struggling with something fundamental. I keep coming across two versions of the hamiltonian for spin orbit coupling: $H_{soc}=\frac{\mu_B}{2c^2}(v \times E) \cdot \sigma $ $\mu_B =$ ...
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1answer
35 views

The MRI signal: why do we consider the phase in the MRI signal

I am trying to understand the imaging principles behind MRI and I was looking at some lecture slides found here Specifically, I am looking at slide 41 where we look at some of the equations regarding ...
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1answer
51 views

Deviation of free falling objects (Coriolis effect) using conservation of angular momentum

I read this pdf on non inertial frame, in particular I have a question on the deviation of free falling object due to Coriolis effect. Consider a ball let go from a tower at height $h$. The ...
2
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1answer
94 views

All the planets in our solar system rotate 'Anticlockwise', except Venus. Why is the only planet that rotate' clockwise'? [duplicate]

All the planets in our solar system rotate 'Anticlockwise', except Venus. Why is the only planet that rotate 'Clockwise'?
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1answer
44 views

Angular momentum consevation and central force

A circular orbit of radius $a$ passing through the centre of a central force is given by the equation $r=2a\cos\theta$. Then using the orbit equation one can show that the force varies as $\vec F(|\...
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0answers
18 views

Wigner-Eckart theorem and Van Vleck paramagnetism

Using the Wigner-Eckart theorem, we can express the matrix elements of Langevin's paramagnetic Hamiltonian $L_z + g_S S_z$ using only the quantum numbers of the total angular momentum, $J$ and $m_J$, ...
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1answer
51 views

How to explain gyroscopic precession in a more intuitional way?

When Studying the phenomenon of precession a classic example installation is this: a wheel spins around the Y axis, gravity applies and the upper part starts spinning around the Z axis. The textbook ...
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18 views

Angular acceleration of rigid body due to a torque

For the rotation of a rigid body about a fixed axis $z$ the following holds. $$ \vec{τ_z}= \frac{d \vec{L_z}} {dt} =I_z \vec{α} \tag{1}$$ Where $ \vec{τ_z}$ is the component parallel to the axis $...
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0answers
37 views

Identical Particle Wavefunctions

Consider the process $$\rho^0\rightarrow\pi^0+\pi^0$$ The $\rho^0$ has $J=1$ whilst the two pions have $S_{tot}=0$ and thus require $L_{tot}=1$ by conservation of angular momentum. Consequently this ...
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1answer
51 views

Contraction of a rotating system

Consider a system of two masses that rotates with constant angular velocity. When a force contracts the system the velocitie of the two masses increase. I understand this in terms of conservation of ...
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1answer
33 views

Uncoupled and coupled bases for electrons in hydrogen atom?

I'm given that for an electron in a hydrogen atom, $L=2$ and $S=1/2$ (quantum numbers associated with $L^2$ and $S^2$). I'm also given that for the uncoupled representation, the basis function is $|L,...
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4answers
65 views

What is the moment of inertia really?

Is moment of inertia or second moment of inertia, simply the resistance of a body to rotate it over an axis? What is radius of gyration? What if the axis is via the center of mass or somewhere ...
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183 views

Coriolis force and conservation of angular momentum

I'm trying to understand the relations between the existance of Coriolis force and the conservation of angular momentum. I found this example on Morin, which confuses me. A carousel rotates ...
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2answers
45 views

Acceleration due to conservation of angular momentum in absence of a force

I don't understand why the conservation of angular momentum can imply an acceleration, in absence of a force. Consider for istance planetary motion. The angular momentum $\vec{L}$ of the planets is ...
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1answer
35 views

why laser beams don't get reflected (or knocked away) when they intersect with each other?

laser beams are photons with the same frequency and the same direction, but according to the wave-particle duality, photons have mass. but if we shoot two masses and they intersect at some point ...
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3answers
54 views

Seeming paradox regarding conservation of angular momentum

Two equal mass astronauts in space are holding hands: one's right hand with the other's left hand and left with right. They are facing each other and are stationary. Astronaut A pulls with her right ...
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58 views

Calculating Clebsch-Gordan coefficients through Racah's formula

So Clebsch-Gordan coefficients are found in tables, but I need to calculate them using Racah's formula, which reads as following: $c_+ (J,M) f_{m_1}^{M+1}=c_+(j_1,m_1-1) f_{m_1-1}^M + c_+(j_2,M-m_1) ...
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1answer
79 views

How to manipulate higher spin systems (higher than spin 1/2) using a given operator?

I’ve been reading ¨Halzen, F., and A. D. Martin. Quarks and Leptons. New York: Wiley Text Books, January 1984. ISBN: 9780471887416¨, and I’d like some clarification of a concept, please: I’m ...
2
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1answer
36 views

Component of angular momentum perpendicular to the rotation axis in rigid body rotation

I have difficulties in understanding, in the rotation of a rigid body, the properties of the component of the angular momentum vector $ \vec {L} $ which is perpendicular to the fixed axis of rotation $...
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0answers
63 views

Rigid body rotation about fixed axis with angular velocity not constant in magnitude

I'm trying to understand the properties of angular momentum in the rotation of a rigid body around a fixed axis $z$, when the angular momentum $\vec{L}$ is not parallel to the angular velocity $\vec{\...
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1answer
50 views

Why will kinetic energy decrease?

Specific situation : A ring of mass M and radius R is rotating about its axis with angular velocity w. Two identical bodies each of mass m are now gently attached at the two ends of a diameter of the ...
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0answers
37 views

Torque on rotating rigid body and pivot point [duplicate]

Consider a rigid body performing a rotational motion around a vertical fixed axis $z$ with constant angular velocity $\vec{\Omega}$. The angular momentum vector $\vec{L}$ is not parallel to $z$ (and ...
1
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2answers
115 views

Forces that exert torque on a rigid body in rotation when angular momentum is not parallel to angular velocity

I'm confused about the rotation of a rigid body, when the angular momentum $\vec{L}$ is not parallel to the angular velocity $\vec{\omega}$. Consider a barbell with two equal masses that rotates ...
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1answer
52 views

Angular momentum, its conservation and Nebula theory

I am carrying out a research work and I am stuck at the first page where there is an introduction to the angular momentum and its relationship in the formation of the solar system. According to that ...
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1answer
71 views

Derivation of Squared Angular Momentum in Spherical Coordinates

While reading my textbook, I found the following: I tried to prove the above equation by doing the following. Knowing that : $$(\vec{A}\times\vec{B}).(\vec{C}\times\vec{D})=(\vec{A}.\vec{C})(\vec{B}...
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1answer
71 views

Mechanics of a falling cat [duplicate]

It is a general statement that cats tend to fall on their feet. How far is this is a valid statement? If I were to drop a cat from 50 meters height, feet up(hypothetically), would it still land on ...
1
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1answer
36 views

Quantum Mechanism Dipole Moment Magnetic Field Problem [closed]

I figured 1a out pretty easily. For 1b I am a bit stuck. So far i have that in the field: $$ \Delta z = v_0t + (1/2) a_zt $$ $$ F_z = ma_z $$ so $$ a_z = F_z/m $$ We know $$ F_z=\mu_z(\partial(B_z)/\...
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0answers
44 views

How can the energy of a spinning black hole be converted into magnetic field?

I've read about the Blandford-Znajek process, and what I understood is that the black hole is treated as if it were a conducting sphere spinning rapidly with an accretion disk around it, and the ...
2
votes
1answer
149 views

A mass hanging under a table: a problem from Goldstein [closed]

I'm trying to solve Problem 1.19 from Goldstein's Chapter 1 (2nd edition), and am getting bogged down in trigonometry (?). Please help me figure out what I'm doing wrong! Two mass points of mass $...
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0answers
27 views

In a spinning spacecraft, the astronaut lets out of a mass attached to a string from a window, what happens to the mass? [closed]

The spacecraft is a cylinder spinning about its cylindrical axis. A mass attached to a long string is slowly let out from the side of the cylinder. Why does it start to perform simple harmonic motion?
2
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0answers
40 views

Is the spin of an electron based on its direction of angular momentum? [duplicate]

Is the spin of an electron based on its direction of angular momentum? Is it based on the right hand rule of angular momentum? Does Spin up means that the electron is spinning anti-clockwise and spin ...
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0answers
31 views

Cylinders rotating on a table

Please consult this video for context, its an interesting watch. https://www.youtube.com/watch?v=JuuYFt8bahE In this video Tadashi spins a tube on a table and it rotates around the tube axis and ...
2
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3answers
60 views

Force at axis of spinning wheel

This is a newbie question, but I don't quite understand the forces at work when an object is rotating. So I've read that i can use my right hand to determine the direction of the force at the axis ...
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0answers
36 views

Statement of vector model of angular momentum in Eisberg and Resnick

While serving as a teaching assistant for a sophomore-level Quantum Physics course, I came across the following paragraph in Eisberg and Resnick regarding orbital angular momentum (section 7-8, page ...
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0answers
65 views

Why are generators defined oppositely in Weinberg's vs. Maggiore's QFT books?

I've been confused about the sign conventions used in Weinberg's QFT book for a long time. Here's my question: The generators $J^{\mu\nu}$ are defined in this book as $$U(1+\omega)=1+\frac{i}{2}\...
12
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3answers
3k views

Could an asteroid land slowly on Earth's surface?

The concept in my mind is that an asteroid is on a vector similar to Earth's, but slightly slower (e.g., 50kmh slower). As Earth passes it, it enters the atmosphere at a sharp angle, and since Earth ...
3
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2answers
63 views

Why do tops move opposite to each other when colliding, not tangentially?

When two well-balanced tops collide, they tend to bounce directly away from each, in other words along the line connecting their centers. Intuitively I would expect the tops to move tangentially, not ...
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3answers
151 views

Conversion of angular momentum to linear momentum in free space

If two objects both with angular and linear velocity collide in free space, can the total linear velocity of the objects increase at the expense of a loss in angular momentum? In other words, imagine ...